How to create a Eye rig with cornea deformation
I tried with the Warp modifier but it seems hard to make it work with a complexe topology, a Lattice modifier seems much more convenient:


First remove all the Copy Transforms constraints of your 3 bones (not sure why you want those).
In Pose mode, close the eyelid a bit so that you'll be able to test the modifier when the eyes are closed.
Create a lattice in front of your eye: shiftA > Lattice. Scale it down.
Subdivide it a bit, for example U and W values > 7, and make it flat on the Y axis with a V value of 1.
Give your eyelid mesh a Lattice modifier. Choose the lattice as Object.
Select your lattice, go in Edit mode, select the central vertices and make it a bit round, use the Proportional Editing / Sphere falloff type to help you.
Parent the lattice to the eye bone: ctrlP > Bone.
Now when you'll move the bone, the transformation of the eyelid will follow.
If necessary make some adjustments to the lattice in Edit mode, or scale it up or down in Object mode, play with the modifier Strength (and create keyframes if the influence changes through time). You can also limit the lattice influence on only a part of the eyelid mesh (with a vertex group that you'll set in the Vertex Group field of the modifier), etc...